Understanding the Adsorption Mechanism of BTPA, DEPA, and DPPA in the Separation of Malachite from Calcite and Quartz: DFT and Experimental Studies

Abstract

The relationship between the structure of bis (2,4,4-trimethylpentyl) phosphinic acid (BTPA), diethyl phosphinic acid (DEPA), and diphenyl phosphinic acid (DPPA) on the flotation performance of malachite was investigated. Through a series of flotation experiments, density functional theory (DFT) calculations, and surface analysis methods, we aimed to deeply understand the microscopic mechanism of the interactions between these collectors and the malachite surface. The experimental results showed that BTPA exhibited excellent selectivity and flotation performance for malachite in the pH range of 5.0–11.0, significantly better than DEPA and DPPA. Surface analysis evidence from X-ray photoelectron spectroscopy (XPS) and Fourier transform infrared spectroscopy (FT-IR) further confirmed the chemical adsorption characteristics of BTPA on the malachite surface. DFT calculations revealed that the adsorption capacity of BTPA on the malachite surface exceeds that of DEPA and DPPA. Electron transfer analysis, especially through frontier molecular orbital theory, differential charge density, PDOS, and COHP analysis, indicated that the charge transfer process from the s orbitals of oxygen atoms in the collectors to the d orbitals of copper atoms on the mineral surface is the decisive factor for the adsorption strength.

1. Introduction

Copper is a crucial resource and raw material for the advancement of modern society [1]. It finds diverse uses across industries such as electric power generation, automotive manufacturing, aerospace engineering, and maritime construction [2,3,4,5,6]. The classification of copper ores primarily consists of copper sulfide minerals and copper oxide minerals. With rapid economic development, the production of copper sulfide minerals has been unable to meet the demands of industrial production [7,8]. Therefore, efficient flotation technology for copper oxide minerals is key to addressing the shortage of copper mineral resources.

Malachite is a common copper oxide mineral, and its flotation techniques mainly involve sulfide flotation and direct flotation [9,10,11]. The sulfide flotation method is primarily used to produce copper sulfide on the surface of malachite by adding sodium sulfide to the slurry, followed by separation through flotation using a collector such as xanthate [12,13]. However, the addition of substantial quantities of sodium sulfide in the actual production process leads to significant environmental pollution. Insufficient or excessive addition of sodium sulfide, resulting from the inexperience of operating workers, can lead to inadequate recovery of malachite during the flotation process [14,15]. Therefore, the separation of malachite by direct flotation has been widely studied.

Choosing the right flotation collector is essential for effectively isolating malachite using direct flotation. Typically, the direct flotation process utilizes collectors like fatty acids [16], hypophosphates [17], and chelating agents [18,19] to segregate malachite from gangue. Through extensive research, it has been observed that fatty acids exhibit limited selectivity in the separation of malachite and calcite. The practical application of chelating agents, such as hydroxamic acids, is hindered by their high cost. The researchers have found that phosphinate exhibits favorable selectivity toward metal ions. Joshua found that selective adsorption on the surface of rare earth minerals can be observed when using phosphinate [20]. The utilization of phosphinate as a collector offers a promising solution to the challenge of effectively separating and enriching Ce-bastnäsite through flotation [21]. Liu found that a-hypophosphite exhibited excellent flotation performance for malachite in comparison to styrene phosphonic acid and sodium isobutylxanthate [17]. Although there have been numerous instances of phosphates being utilized in the flotation of metal oxide ores, developing an effective phosphate for copper oxide flotation still poses a significant challenge.

In our study, the phosphinate-based collectors BTPA, DEPA, and DPPA were chosen based on their chemical compatibility and performance profiles observed in preliminary tests. BTPA was found to be particularly effective in enhancing the flotation recovery and grade of malachite under controlled experimental conditions. In contrast, DEPA and DPPA did not facilitate the flotation of any oxide minerals under similar conditions, highlighting the critical need for selecting the right collector based on specific ore characteristics and flotation environments. The flotation performance of malachite, quartz, and calcite has been extensively evaluated by BTPA, DEPA, and DPPA. The micro-flotation tests were used to examine how BTPA behaves in flotation in conjunction with X-ray photoelectron spectroscopy (XPS) and Fourier transform infrared spectroscopy (FTIR). Furthermore, density functional theory (DFT) was employed to scrutinize the molecular compositions of BTPA, DEPA, and DPPA, as well as their electron transfer processes during absorption onto the malachite interface.

2. Materials and Methods

2.1. Minerals and Reagents

The malachite used in this project was sourced from Guangdong Province, China, while the calcite and quartz were obtained from Hunan Province, China. All samples were crushed using zirconium balls. After crushing, particles ranging in size from 38 μm to 74 μm were collected for flotation tests and other analyses. Figure 1 displays the XRD spectra of malachite, calcite, and quartz; it is evident that these spectra are consistent with those reported in the literature [22,23,24].

Table 1. Multi-element assay of the malachite, calcite, and quartz samples (%).

Sample	Cu	Al	Fe	Ca	Mg	SiO2	C
Malachite	55.89	0.04	0.21	1.97	0.05	0.52	6.14
Calcite	0.03	0.07	0.048	39.39	0.27	0.29	11.28
Quartz	0.05	0.04	0.04	2.06	0.03	97.62	0.02

presents the multi-element analysis data for the three mineral samples. As shown in Table 1, the purity of malachite, calcite, and quartz are, respectively, determined as 98.2%, 97.6%, and 98.5%. The three collectors (BTPA, DEPA, and DPPA) used in the flotation tests were obtained from Aladdin Scientific. Sodium hydroxide (NaOH) and hydrochloric acid (HCl), used for adjusting the pH of the slurry, were sourced from Sinopharm.

2.2. Micro-Flotation Experiments

In our study, the flotation test for pure minerals such as malachite, calcite, and quartz was conducted using a 30 mL XFGII-type flotation machine, which operated at a constant speed of 1758 revolutions per minute without the use of external airflow. This ensured sufficient agitation to distribute the collector evenly and maintain the particles in suspension, which is crucial for the flotation process. The entire procedure was carried out in the same cell for both the conditioning of the minerals with collectors and the micro-flotation tests, thus maintaining consistency in sample treatment and test conditions. Initially, 2.0 g of the mineral sample is weighed and introduced into the flotation cell, which already contains 30 mL of a pre-determined solution. The solution’s pH is adjusted to the required level between 5 and 12 using either NaOH or HCl, and the system is stirred for two minutes. After this, the selected collector is added, followed by additional stirring for another two minutes to ensure thorough mixing. The flotation process itself lasts four minutes, resulting in the separation of concentrate and tailings. The masses of the dried concentrate and tailings are recorded, and the flotation recovery rate is calculated using Equation (1).

For the artificial mixed ore flotation tests involving malachite and calcite (or quartz), the methodology includes measuring out 1.0 g of each mineral and placing them into the same 30 mL flotation cell. After the pH adjustment and initial stirring, the collector is introduced, and the mixture is stirred again for another two minutes. A four-minute flotation test follows, leading to the separation of concentrate from the tailings, with their respective weights noted for recovery calculation according to Equation (2). Figure 2 visually illustrates the flotation procedure outlined above, providing a comprehensive view of the experimental setup and the sequence of operations performed.

$${\epsilon }_p=\frac{m_f}{m_f+m_t}$$

(1)

$${\epsilon }_{M/C/Q}=\frac{m_f\times C_1}{m_f\times C_1+m_t\times C_2}$$

(2)

where ε_p represents the flotation recovery of malachite, calcite, or quartz in their pure mineral flotation test. The m_f denotes the mass of concentrate obtained from froth flotation, while m_t refers to the mass of tailing collected from the slurry in Equation (1). In Equation (2), ε_M/C/Q represent the recoveries of malachite, calcite, and quartz, respectively, in an artificial mixed mineral flotation test. The m_f signifies the mass of concentrate obtained from froth flotation of the artificial mixed mineral, and mt indicates the mass of tailings collected from its slurry. C₁ and C₂ denote the grades of minerals present in concentrates and tailings correspondingly.

2.3. XPS Measurements

The research used a Thermo Scientific K-Alpha device, sourced from Thermo Fisher Scientific, located in Waltham, MA, USA. This device utilized monochromatic Al Kα radiation for X-ray Photoelectron Spectroscopy (XPS) analysis, enabling a precise and reliable assessment of the surface chemical composition of the materials. The tests were conducted in a high vacuum environment (pressure under 1.0 × 10⁻⁹ Torr), accompanied by an extensive energy range measurement (0 eV to 1200 eV), with each sample undergoing no less than three repeated scans to validate the accuracy and consistency of the findings. Additionally, adjusting the spectrometer using the binding energy (B.E.) further improved the precision of the results. The C 1s peak was calibrated at 284.8 eV [25,26,27].

2.4. FT-IR Spectra Analysis

The Fourier Transform Infrared (FT-IR) spectra of malachite, BTPA, and malachite adsorbed on the surface by BTPA were measured using the KBr disc method with a Nicolet IS20 instrument. The infrared spectra used for the measurements were in the wavelength range of 400–4000 cm⁻¹.

2.5. Computation Approach

This study uses the Dmol3 model in Materials Studio software, version 2017, to perform a series of density functional theory (DFT) calculations. These calculations aimed at an in-depth exploration of the adsorption characteristics of the three collector molecules and their interaction with the surface of malachite. The choice of the Generalized Gradient Approximation (GGA) method, finely tuned using the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional, was crucial in guaranteeing the precision and reliability of the findings [28,29,30].

In order to enhance the crystal unit cells and reagent molecules, strict convergence criteria were enforced. These criteria consisted of an energy alteration of 1.0 × 10⁻⁵ Ha, a highest force of 2.0 × 10⁻³ Ha·Å⁻¹, a maximum displacement of 5.0 × 10⁻³ Å, and a maximum step size and self-consistent field adjusted to 1.0 × 10⁻⁶ eV/atom. These criteria ensured the stability of the calculation process and the precision of the outcomes. The calculation incorporated the contributions of all electrons, setting the kinetic energy cutoff to 351 eV to boost computational efficiency and accuracy. In order to enhance the credibility of the computational findings, a k-point grid of 2 × 2 × 6 was utilized for structural calculations. Moreover, to simulate the weak interactions between molecules more precisely, we introduced DFT-D correction by employing the TS method.

The crystal structure data for malachite was obtained from a previous study in crystallography [31]. Based on previous studies, this paper utilizes the (−2 0 1) face of malachite to investigate its interaction with the collector due to its superior thermodynamic stability [32,33].

The (2 × 2) malachite (−2 0 1) surface supercell, which consists of five atomic layers and includes a vacuum spacing of 25 Å, was constructed to explore the equilibrium geometries of the malachite (−2 0 1) surface. The optimization of collector molecules is carried out at the gamma point within a cubic box measuring 15 × 15 × 15 ų. The resultant equilibrium geometries of the malachite (−2 0 1) surface are illustrated in Figure 3, providing a clear visualization of the optimized structure.

The configurations of valence electrons for Cu, C, P, S, H, and O are 3d¹⁰4s¹, 2s²2p², 3s²3p³, 3s²3p⁴, 1s¹, and 2s²2p⁴, respectively.

Use the adsorption energy determined by Equation (3) to evaluate the adsorption capacity of the flotation agent on the mineral surface.

$$E_{ads}=E_{adsorbates/surface}-E_{surface}-E_{adsorbates}$$

(3)

Among them, E_ads is the adsorption energy, E_{adsorbates/surface} is the calculated energy of the flotation agent adsorbed on the malachite surface, E_surface is the surface energy of malachite, and E_adsorbates is the energy of the flotation agent.

3. Results and Discussion

3.1. Micro-Flotation Experiments for Single Mineral

Figure 4 illustrates the flotation recovery rates when utilizing DEPA and DPPA as collectors within the pH range of 6–11. The results show that recovery rates for malachite, quartz, and calcite approach zero, indicating minimal adsorption of these minerals by DEPA and DPPA. Conversely, when using BTPA as the collector and increasing the pH from 6 to 10, the recovery rate of malachite notably increased from 77.10% to 91.50%, demonstrating BTPA’s effectiveness. However, at pH 11, the recovery rate slightly decreased to 88.85%. Similarly, as the pH and BTPA dosage increased, the recovery rates of quartz and calcite decreased significantly, with quartz dropping from 65.46% to 4.43% and calcite from 73.93% to 12.57%.

The micro-flotation experiments were conducted in duplicate to ensure accuracy and reproducibility. This rigorous approach allowed us to maintain the variability in recovery rates within a ±5% error range for both pure and mixed mineral experiments, reflecting high reproducibility. The figures now display average values with corresponding error bars to better illustrate the variability and reliability of our data. This detailed presentation confirms that within the pH range of 6–11, BTPA acts as a highly selective collector for malachite, particularly achieving its peak recovery rate at pH 10. In contrast, DEPA and DPPA exhibit very low adsorption for these minerals within this pH range, almost not affecting their flotation recovery rates.

In the experimental study shown in Figure 5, the effect of three receptors (BTPA, DEPA, and DPPA) on the flotation efficiency of malachite, quartz, and calcite at pH 10 was studied. The results show that BTPA significantly improves the flotation efficiency of malachite. The rate surged significantly from 19.25% at 1 mg/L to 90.67% at 5 mg/L, marking the most pronounced enhancement. Conversely, the increase in quartz recovery rate was comparatively modest, elevating from 6.37% to 30.11%. DEPA and DPPA, however, exhibited minimal impact on the flotation recovery rates of the three minerals.

3.2. Micro-Flotation Separation Results for Artificially Mixed Minerals

This study investigates the effects of BPTA on the flotation recovery and grade of malachite and calcite, as well as on composite ore samples containing both malachite and quartz, conducted at a controlled pH of approximately 10. The micro-flotation experiments were performed in duplicate to ensure the accuracy and reproducibility of results, with each set of tests demonstrating consistent improvements in the recovery and grade of malachite. With increased dosages of BPTA, recovery rates exceeded 80%, and grade levels surpassed 60%, as depicted in Figure 6a, where error bars represent a ±5% error range, illustrating the high reproducibility of our results. For calcite, although the recovery rates increased both before and after a dosage of 3 mg/L, they remained below 40%. The grade of calcite showed minimal variation, consistently staying below 20%.

Further experimentation identified the optimal BPTA dosage at 3.0 mg/L, under which conditions malachite achieved its highest recovery and grade values of 91.45% and 64.25%, respectively. Conversely, the recovery and grade for calcite decreased significantly to 26.39% and 13.66%, respectively. The experiment also highlighted that the optimal flotation separation effect was achieved when the pH ranged from 9.0 to 10.0, with malachite recovery and grade exceeding 93.82% and 62.88%, respectively, as illustrated in Figure 6b. However, calcite recovery and grade diminished to below 26.17% and 14.52%, respectively.

When further investigating the flotation separation of malachite and quartz in the mixed mineral sample, the recovery rate and grade remained stable. With the BPTA dosage increased to 3.0 mg/L, the peak values reached 93.82% and 62.88%, respectively, as shown in Figure 7a. By adjusting the pH to a range of 8.0–9.0, optimal separation conditions for malachite were achieved, with recovery and grade exceeding 92.83% and 68.94%, as shown in Figure 7b. Nonetheless, the quartz yield and grade dropped significantly, remaining lower than 20.0% and 10.0%, respectively.

3.3. XPS Analysis

Figure 8a shows the XPS measurement results of malachite before and after BPTA treatment. On the surface of malachite, the P 2p XPS peak is observed, indicating chemical changes during the treatment process. Further insights were gained through high-resolution XPS spectra analysis, which detailed the characteristics of Cu2p, P2p, and O1s as depicted in Figure 8a–c. The Cu 2p_2/3 XPS peak of the untreated malachite appears at approximately 954.77 eV, whereas the Cu 2p1/2 XPS peak appears at approximately 934.71 eV [34]. However, after BPTA treatment, The Cu 2p_2/3 XPS peak was observed at 954.45 eV, while the Cu 2p_1/2 XPS peak shifted to 934.43 eV, indicating electron transfer from the -OH group to copper atoms, reflecting the charge transfer phenomenon during the treatment process. Further, P 2p XPS analysis (Figure 8c) revealed no significant peaks between 140 and 125 eV, implying the unique nature of the P 2p signal. However, upon BPTA adsorption onto the malachite surface, a P 2p XPS peak appears at 132.39 eV [35,36], confirming the successful chemical adsorption of BPTA. Figure 8d presents the O 1s XPS analysis results. The O 1s XPS spectrum of unmodified malachite displays two peaks corresponding to carbonate ions CO₃²⁻ at 531.08 eV and hydroxide ions (OH-) at 531.94 eV. Upon BPTA adsorption, a new O 1s XPS peak appears at 529.11 eV, indicating an O-Cu-O bond and suggesting successful BPTA adsorption. In conclusion, BPTA successfully adsorbs onto the surface of malachite.

3.4. FT-IR Spectra Analysis

As shown in Figure 9, FT-IR spectroscopic analysis revealed the spectral characteristics of three samples: BPTA, malachite, and malachite treated with BPTA. The main peaks identified in the FT-IR spectrum of malachite corresponded to the presence of (CO₃)²⁻ and -OH groups. Specifically, the peaks at approximately 3404 cm⁻¹ and 3311 cm⁻¹ were clear indicators of the stretching vibrations of the -OH group, while the peaks around 1501 cm⁻¹ and 1384 cm⁻¹ represented the asymmetric stretching vibrations of the (CO₃)²⁻ group.

During the FT-IR analysis of BPTA, numerous important peaks were detected. The peaks at 3408 cm⁻¹ and 3315 cm⁻¹ were associated with the stretching vibrations of the -OH group, while those at 2956 cm⁻¹ and 2902 cm⁻¹ corresponded to the stretching vibrations of the -CH₂ and -CH₃ groups, respectively. Additionally, the peaks at 1507 cm⁻¹ and 1388 cm⁻¹ represented the vibrations of P=O and P-O-H bonds [37,38,39], and the peak near 1155 cm⁻¹ indicated the presence of a P-C absorption band.

FT-IR spectral analysis of malachite treated with BPTA showed clear C-H absorption bands around 2953 cm⁻¹ and 2858 cm⁻¹, along with a likely O-H vibration peak near 1678 cm⁻¹. Considering the strong infrared absorption band characteristics of malachite itself, the unique absorption peaks of the Cu-BPTA surface complex appeared relatively subdued. This observation suggests the impact of BPTA treatment on the surface properties of malachite and the potential interactions occurring during the formation of the Cu-BPTA surface complex [40].

3.5. Molecular Structure and Electronic Structure of Collector and Depressant

Figure 10 presents the geometrically optimized structures and front orbital diagrams of the three collector ions. According to first-principles calculations, the P-O bond length from BTPA is 1.525 Å, which is slightly longer than that from DEPA (1.521 Å). This indicates that BTPA is more reactive than DEPA. Conversely, DPPA has the shortest P-O bond at 1.514 Å, indicating that it is the least reactive among the three collector ions.

Table 2. HOMO, LUMO energy, and Mulliken charge of BTPA, DEPA, and DPPA.

Reagent	HOMO/Ha	LUMO/Ha	Atomic Charge (|e−|)
			P	O1	O2
BPTA	−0.01729	0.17269	1.152	−0.762	−0.756
DEPA	−0.03287	0.11803	1.180	−0.751	−0.749
DPPA	−0.04361	0.06258	1.197	−0.737	−0.734

provides a breakdown of the charges on the P and O atoms present in the collector ions. The P atom is positively charged, while the O atom holds a negative charge. DPPA shows a lower negative charge on its O atoms in comparison to other collectors. Additionally, BTPA displays the most substantially negative charge on its O atoms, indicating increased reactivity and stronger adsorption onto the malachite surface.

Figure 10 illustrates the positioning of the Highest Occupied Molecular Orbital (HOMO) isosurfaces primarily on the oxygen atoms and the Lowest Unoccupied Molecular Orbital (LUMO) isosurfaces predominantly on the carbon chains in the three collectors [41]. The HOMO regions act as electron donors, while the LUMO regions serve as electron acceptors. These energy levels of the HOMO and LUMO demonstrate the reactivity of the collectors before adsorption. According to the information presented in Table 2, BTPA exhibits not only the highest electron donation capacity but also electron acceptance capacity, as demonstrated by its HOMO values ranking highest in the order of BTPA > DEPA > DPPA, with the LUMO values showing a similar trend.

3.6. Geometries and Adsorption Energies of Collectors on the Malachite (−2 0 1) Surface

This study investigates the adsorption characteristics of phosphonic acid collectors (BTPA, DEPA, DPPA) on malachite (−2 0 1) surfaces, as illustrated in Figure 11. In particular, this study focuses on the Cu-O bond distances between the collectors and the malachite surface and monitors the alterations in the P-O bond distances within the collectors before and after adsorption, as outlined in

Table 3. Relaxed bond lengths (in Å) for BTPA, DEPA, and DPPAC collectors adsorption on malachite (−2 0 1) surface.

Bonds/Å	BTPA	DEPA	DPPA
After adsorption
Cu1-O1	2.019	2.057	2.075
Cu2-O2	2.198	2.218	2.382
P-O1	1.533	1.537	1.532
P-O2	1.518	1.518	1.515
Before adsorption
P-O1	1.521	1.522	1.514
P-O2	1.517	1.519	1.515

. This investigation uncovers the intricate microscopic processes by which these collectors engage with malachite surfaces and influence variations in bond distances.

Among the three phosphonic acid collectors, the Cu-O bonds formed by BTPA (with respective lengths of 2.019 Å and 2.198 Å) are shorter than those formed by DEPA, whereas the Cu-O bonds formed by DPPA (with respective lengths of 2.075 Å and 2.382 Å) are the longest. These observations suggest that BTPA exhibits stronger adsorption ability on malachite surfaces than DEPA.

As delineated in Table 3, the lengths of P-O1 bonds for all three collectors increased (from 1.521 Å, 1.522 Å, and 1.514 Å to 1.533 Å, 1.537 Å, and 1.532 Å, respectively), indicating a weakening of P-O1 bonds due to the adsorption of the collectors on the malachite surface. Concurrently, the lengths of P-O2 bonds for all three collectors remained nearly unchanged before and after adsorption, suggesting minimal chemical bond formation between Cu and O2.

The importance of adsorption energy lies in its ability to indicate the strength of adsorption of flotation agents on mineral surfaces, a key factor in optimizing the flotation process during mineral processing. A greater adsorption energy indicates improved adsorption capacity.

Table 4. Simulated adsorption energies of three collectors.

Etotal/(kJ/mol)	Emolecular/(kJ/mol)	Esurface/(kJ/mol)	Eadsorption/(kJ/mol)	Etotal/(kJ/mol)
BTPA	−11507858.08	−418484.1257	−11087894.09	−1479.863324
DEPA	−11503136.28	−414938.3767	−11087894.09	−303.8134923
DPPA	−11518147.93	−429992.0345	−11087894.09	−261.8131775

illustrates the variations in adsorption energies among three distinct collectors on malachite, demonstrating that DPPA exhibits the least adsorption energy at −261.81 kJ/mol. This implies a lower adsorption capacity, indicating limited efficacy in mineral flotation operations. In stark contrast, BTPA demonstrates a substantially higher adsorption energy of −1479.86 kJ/mol, underscoring its potent adsorption capability and suggesting its potential for superior flotation performance. These observations are corroborated by the results of flotation experiments, which align with the varying adsorption energies, thereby reinforcing the critical role of adsorption energy in influencing flotation efficacy.

Figure 12 illustrates the distinct charge density of the three collectors found on the malachite surface. Charge accumulation is denoted by the blue regions, while charge depletion is shown in yellow areas. This visualization validates the occurrence of charge transfer via Cu-O bonds between the oxygen atoms of the collectors and the copper atoms on the malachite surface, highlighting the importance of this process for adsorption. Comparative analysis of these collectors’ adsorption shows a notable increase in charge density in the Cu1-O1 bonds and a decrease in the P-O1 bonds, indicating a weakening of the P-O1 bond concurrent with the strengthening of the Cu1-O1 bond. No notable change in the charge density of Cu2-O2 bonds was detected, implying the absence of chemical bond formation between Cu2 and O2. These observations are consistent with the front orbital analysis previously discussed in Section 3.5, suggesting that the propensity for electron donation by the collectors enhances their adsorption on malachite surfaces.

Partial Density of States (PDOS) and Crystal Orbital Hamilton Population (COHP) of the three collectors adsorbed on the malachite surface were studied to analyze the adsorption behavior of the collector molecules on the mineral surface [42,43]. The changes in fractional densities of all three collectors before and after adsorption on the mineral surface exhibit a consistent trend (Figure 13). Upon adsorption, the s orbitals of the oxygen atoms in these collectors penetrate further into the valence band. The analysis of COHP reveals a prominent peak (S1) in the range of −0.8 to −0.6 Ha, indicating an interaction between the s orbitals of oxygen atoms within the collectors and the Cu atom orbitals present on the malachite surface. The area of overlap between the p orbitals of oxygen atoms in collectors and the d orbitals of Cu atoms on malachite primarily occurs in the energy range of −0.4 to 0 Ha. This points towards an interaction between the p orbitals of oxygen atoms and d orbitals of Cu atoms at this particular energy level. Additionally, the COHP analysis displays peaks (S2) both above and below the 0-axis within this energy range, suggesting the presence of both bonding and antibonding interactions between d orbitals of Cu atoms on the malachite surface and p orbitals of O atoms.

In

Table 5. This COHP of BTPA, DEPA, and DPPA to Cu atom.

Collector	Bond	Peak	Aera
BTPA	Cu1-O1	S1	−0.03018
		S2	−0.02145
DEPA	Cu1-O1	S1	−0.02585
		S2	−0.02028
DPPA	Cu1-O1	S1	−0.02463
		S2	−0.02099

, through the analysis of the adsorption characteristics of three different collectors (BTPA, DEPA, DPPA) on the surface of malachite, subtle differences in their chemical interactions were revealed. The size of the S2 region (as seen in Table 5) suggests that these three collectors have roughly the same interaction strength between the p orbitals of O atoms and the d orbitals of Cu atoms, indicating a similar mechanism of action in this particular aspect. However, the significant advantage of BTPA in the S1 region reveals its stronger bond strength with the malachite surface, suggesting superior adsorption performance of BTPA.

This conclusion is further reinforced by the fact that the strength of the Cu1-O1 bond formed by BTPA surpasses that of DEPA and DPPA, especially when compared to DPPA, which forms the weakest Cu1-O1 bond. This difference highlights the key factor influencing the strength of collector adsorption on the malachite surface: the bonding between the s orbitals of O atoms and the d orbitals of Cu atoms on the malachite surface.

In order to provide additional insight into how the molecular arrangement of the collector impacts its ability to adsorb on the malachite surface, an examination was carried out on the transfer of electronic charges throughout the adsorption process. The level of charge exchange between the collector and the mineral surface acts as a gauge of their adsorption capability. Mulliken charge analysis revealed that BTPA and DEPA lost 0.08e and 0.02e, respectively, upon adsorption to malachite, indicating electron transfer from these collectors to the mineral surface. Furthermore, BTPA exhibited a greater capacity to transfer electrons to malachite compared to DEPA. DPPA exhibited an adsorption gain of 0.04e on the malachite surface, indicating that malachite functions as an electron donor while DPPA acts as an electron acceptor during their interaction on the mineral surface.

Table 6. Mulliken atomic charges (|e⁻|) of BTPA, DEPA, and DPPA collector O1 and Cu atoms of malachite (−2 0 1) surface, before and adsorption.

	Adsorption State	Atom	s	p	d	Total	Charge	Δ Charge
BTPA DEPA	Before	O1	1.92	5.00	0	6.92	−0.92	−0.05
	After		1.88	5.09	0	6.97	−0.97
	Before	Cu1	0.46	0.15	9.46	10.08	0.92	0.13
	After		0.40	0.17	9.38	9.95	1.05
	Before	O2	1.92	4.95	0	6.87	−0.87	−0.07
	After		1.87	5.07	0	6.94	−0.94
	Before	Cu2	0.46	0.18	9.52	10.15	0.85	0.19
	After		0.42	0.18	9.38	9.98	1.04
DEPA	Before	O1	1.92	4.97	0	6.89	−0.89	−0.07
	After		1.88	5.07	0	6.96	−0.96
	Before	Cu1	0.46	0.15	9.46	10.08	0.92	0.11
	After		0.41	0.18	9.37	9.97	1.03
	Before	O2	1.92	4.95	0	6.87	−0.87	−0.07
	After		1.88	5.06	0	6.94	−0.94
	Before	Cu2	0.46	0.18	9.52	10.15	0.85	0.17
	After		0.41	0.16	9.40	9.96	1.02
DPPA	Before	O1	1.92	4.99	0	6.90	−0.90	−0.07
	After		1.87	5.10	0	6.97	−0.97
	Before	Cu1	0.46	0.15	9.46	10.08	0.92	0.09
	After		0.45	0.12	9.42	9.99	1.01
	Before	O2	1.92	4.97	0	6.89	−0.89	−0.09
	After		1.88	5.09	0	6.98	−0.98
	Before	Cu2	0.46	0.18	9.52	10.15	0.85	0.19
	After		0.42	0.18	9.37	9.96	1.04

shows that oxygen atoms in all three collectors lose electrons in the s orbitals while gaining electrons in the p orbitals. Similarly, copper atoms on the malachite surface lose electrons in the s and d orbitals while gaining electrons in the p orbitals. These results indicate that both electron transfers from collectors to mineral surfaces and vice versa occur during their adsorption on malachite.

3.7. Discussion

This study revealed differences in the adsorption capabilities of three phosphonic acid collectors on the surface of malachite, primarily due to variations in their carbon chain structures. Research has shown that the adsorption capacity of mineral surface collectors is related to their electron transfer capacity. As confirmed by Zheng et al. [22], phosphonic acid collectors exhibit excellent flotation performance on malachite against calcite and quartz due to the electron donation from other conjugated structures. Thus, the electron transfer capability of the collectors plays a crucial role in influencing their adsorption capacity.

Our Density Functional Theory (DFT) calculations further detailed how the carbon chain structure is a key factor affecting the Highest Occupied Molecular Orbital (HOMO) of the collectors. The higher the HOMO energy, the stronger the electron-donating activity of the collectors. The sequence of HOMO energies for the three collectors is BTPA > DEPA > DPPA. Differential charge density analysis demonstrated that there was charge transfer between the oxygen atoms of the collectors and the active copper atoms on the mineral surface, facilitating chemical adsorption. Crystal Orbital Hamilton Population (COHP) analysis showed that when the oxygen atom’s p orbitals bond with the copper atom’s d orbitals on the mineral surface, electrons occupy both bonding and antibonding orbitals. The COHP values for the interaction between the oxygen atom’s p orbitals of the collectors and the copper atom’s d orbitals on the mineral surface were roughly similar among the three collectors. Additionally, bonding orbitals formed between the oxygen atom’s s orbitals and the copper atom’s d orbitals on the mineral surface. Consistent with Li et al.’s findings, the headgroup’s O or N heteroatoms act as the centers for electron donation, i.e., the centers for chemical reactions [18]. The strength of the interaction between these orbitals follows the sequence BTPA > DEPA > DPPA, which aligns with the adsorption energy sequence.

Charge transfer analysis further confirmed the significance of electron transfer during chemical adsorption. Both differential charge density analysis and Mulliken charge analysis revealed a bidirectional charge transfer mechanism between the oxygen atoms of the collectors and the copper atoms on the mineral surface. The sequence of charge transfer from the collectors to the mineral surface was BTPA > DEPA > DPPA, consistent with the sequence of HOMO values and adsorption energies. The overall electron transfer interactions between BTPA, DEPA, and DPPA collectors and malachite are illustrated in Figure 14.

Therefore, the strength of the interaction between the oxygen atom’s orbitals of the phosphonic acid collectors and the copper d orbitals on the mineral surface is crucial for the adsorption capacity of the collectors. Modifying the carbon chain structure can alter the electron transfer from the oxygen atom orbitals of the collectors to the active sites on the mineral surface, consequently affecting the adsorption of the collectors on malachite. This guides the molecular design of phosphonic acid collectors.

4. Conclusions

Flotation technology is a core technique in the field of mineral processing, especially significant in the separation of economically valuable minerals, such as malachite, from associated minerals like calcite and quartz. Comparative analysis of micro-flotation experiments revealed that among the collectors BTPA, DEPA, and DPPA, BTPA demonstrates notable selectivity and collecting capacity for malachite within the pH range of 5.0–11.0. The discovery underscores the significant benefit of BTPA as a reagent in the malachite flotation process, predominantly due to its unique chemical affinity for the malachite surface.

Additional data obtained from Fourier Transform Infrared Spectroscopy (FT-IR) and X-ray Photoelectron Spectroscopy (XPS) examinations support the conclusion that BTPA creates a stable adsorption film on the malachite surface via chemical bonds, a critical factor in its outstanding flotation capabilities. In addition, a comparative analysis of the electron-donating abilities further confirmed that BTPA can more effectively engage in electron exchange with copper (Cu) atoms on the surface of malachite, significantly enhancing its flotation effect.

Regarding the adsorption mechanism, the copper–oxygen (Cu-O) bond formed between the collector and the copper atoms on the surface of malachite is achieved through charge transfer and interaction between the s orbitals of the collector’s oxygen atoms and the d orbitals of the surface copper atoms of malachite. In this process, BTPA showed the strongest adsorption capability, further validating its superior performance as a collector in the flotation process of malachite.

These findings provide crucial theoretical support for the development of more efficient malachite collectors, underscoring the importance of a deep understanding of the mechanisms of interaction between collectors and mineral surfaces. This is not only vital for enhancing the efficiency and selectivity of flotation technology but also has a long-term impact on advancing and optimizing mineral processing techniques.